Total No. of Pages (including this): 7

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EE 131A Midterm

Wednesday November 13, 2013

DO NOT OPEN UNTIL YOU ARE TOLD TO DO SO

You have 1 hr. and 50 min.

Only this booklet and Two sheets of notes should be on your desk

Lecture Notes, Homework solutions, and any books are not allowed

Calculators maybe used if desired.

Write your answer to each question in the space provided

Read the problem statements carefully Show your work and reasoning clearly

For partial credit justify your results:

Simply writing down one line with the correct answer is not adequate

Your Name and Student Id#:

Name and Student Id:

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Problem 1. (total: 25pts) In a game of poker five cards are picked at random from a deck of 52 cards. Note that a deck of cards has 13 denominations (or kinds) (namely, Ace, 2, 3, 4, 5, 6, 7, 8, 9, 10, Jack, Queen, and King) and 4 suits (namely, Hearts, Spades, Clubs, and Diamonds). What is the probability of being dealt

(a) (5pts) A flush? That is, all five cards are from the same suit. So, Ace, 10, Jack, Queen and King of Spades will comprise a flush.

(b) (5pts) One pair? This occurs when the cards have denominations a, a, b, c, d, where a, b, c, and d, are all distinct. So 10 of Spades, 10 of Hearts, 2 of Clubs, 3 of Diamonds, and Jack of Diamonds will comprise One Pair.

(c) (5pts) Two Pairs? This occurs when the cards have denominations a, a, b, b,

c, where a, b, and c, are all distinct. So 10 of Spades, 10 of Hearts, 2 of Clubs, 2 of Hearts, and Jack of Diamonds will comprise Two Pair.

(d) (5pts) Three of a kind? This occurs when the cards have denominations a, a,

a, b, c, where a, b, and c, are all distinct. So 10 of Spades, 10 of Hearts, 10 of Clubs, 2 of clubs, and Jack of Diamonds will comprise Three of a kind.

(e) (5pts) Four of a kind? This occurs when the cards have denominations a, a,

a, a, b, where a and b, are distinct. So 10 of Spades, 10 of Hearts, 10 of Clubs, 10 of Diamonds, and Jack will comprise Four of a kind.

Name and Student Id:

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Name and Student Id:

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Problem 2 (total: 20 pts) A woman has n distinct keys, of which only one will open her door.

(a) (10 pts) If she tries the keys at random, each time discarding those that do not work, what is the probability that she will open the door on her kth try?

(b) (10 pts) If she does not discard the previously tried keys (that is, each time she tries a

randomly picked key from the entire set of n keys.) then what is the probability that she will open the door on her kth try?

Name and Student Id:

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Problem 3. (20pts) A worker has asked her supervisor for a letter of recommendation for a new job. She estimates that (i) with probability 0.8 she will get the job if she receives a Strong recommendation, (ii) with probability 0.4 she will get the job if she receives a Moderate recommendation, and (iii) with probability 0.1 she will get the job if she received a Weak recommendation. She further estimates that the probabilities the recommendation will be Strong, Moderate, or Weak are 0.7, 0.2 and 0.1, respectively.

a. (10 pts) What is the probability that she will receive a the new job offer?

b. (5 pts) Given that she does receive the offer, what is the probablility that she received a Strong recommendation?

c. (5 pts) Given that she does not receive the offer, what is the probability that she received

a Weak recommendation?

Name and Student Id:

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Problem 4. (total: 20 pts) A gambling book recommends the following winning strategy for the game of roulette. It recommends that a gambler bet $1 on Red. If Red appears (which has probability of 18/38) then the gambler should take her $1 profit and quit. If the gambler loses this bet (which has probability 20/38 of occurring), she should make additional $1 bets on red on each of the next two spins of the roulette wheel and the quit. Let X be the random variable that denotes the gamblers winnings when she quits.

a. (5 pts) State all the outcomes in the underlying sample space in terms of wins and losses. For example, if she loses her first bet, but wins the two next ones, then the outcome is LWW.

b. (10 pts) What are the values that the random variable X can take and what are their

respective probabilities? What is P[X>0]?

c. (5 pts) What is E[X]? Is this a winning strategy? Note 1: To answer this question, you do not need to know anything more about the game of roulette than the information provided here. Note 2: When the player bets $1 on Red, if Red appears then she makes a profit of $1, i.e., her winning from this bet is +$1. If Red does not appear, then she loses the $1 she bet, i.e., her winning from this bet is -$1.

Name and Student Id:

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Problem 5. (total: 15 pts)

An LCD display has 1000X750 pixels. A display is accepted if it has 15 or fewer faulty pixels. The probability that a pixel is faulty coming out of the production line is 10-5. Using the Poisson approximation find the proportion of displays that are accepted.